Coffee mug car holder, tilting to spill.

In my car, a Dodge Grand Voyager minivan, there is a coffee cup holder, it has a spring mounted side clip that would hold it firmly but it is broken and so extends fully out like a little wing and does not hold the cup well.

So my coffee cup I get from WaWa holds 600 ml of liquid, the top ID is 85 mm wide and the bottom ID is 60 mm wide, its total 150 mm high but the bottom is elevated by 10 mm so there is a gap at the bottom.

The lip of the coffee cup holder supports the cup 40 mm from the bottom, or in other words, 30 mm of liquid is possible below the top of the lip support.

I see that when I accelerate, if I don't gun it, the liquid stays in the cup, there is a lid on top to keep coffee from spilling.
But if I really step on the pedal, the whole think falls out.

So here is the question: considering there will be different heights of liquid in the cup after drinking some of it, it would appear intuitively obvious the lower the liquid level, the more I can accelerate and not lose the cup out of the holder. So what is the optimum height of liquid to allow the maximum amount of acceleration? If the height goes to zero then it would seem the amount of acceleration would also go down because the cup only weighs about 2 grams and 110 mm sticks up past the lip, so it seems to me there should be an optimum liquid level to allow the maximum acceleration. What is that level?

Originally posted by sonhouse
In my car, a Dodge Grand Voyager minivan, there is a coffee cup holder, it has a spring mounted side clip that would hold it firmly but it is broken and so extends fully out like a little wing and does not hold the cup well.

So my coffee cup I get from WaWa holds 600 ml of liquid, the top ID is 85 mm wide and the bottom ID is 60 mm wide, its total 150 mm ...[text shortened]... there should be an optimum liquid level to allow the maximum acceleration. What is that level?

If the height goes to zero then it would seem the amount of acceleration would also go down because the cup only weighs about 2 grams and 110 mm sticks up past the lip, so it seems to me there should be an optimum liquid level to allow the maximum acceleration. What is that level?

I don't understand. It seems obvious that the lower the liquid the more you can accelerate...

Unless you're suggesting the liquid will lower the center of gravity of the cup, stabilising it. Is that what you mean?

Originally posted by AThousandYoung
[b]If the height goes to zero then it would seem the amount of acceleration would also go down because the cup only weighs about 2 grams and 110 mm sticks up past the lip, so it seems to me there should be an optimum liquid level to allow the maximum acceleration. What is that level?

I don't understand. It seems obvious that the lower the liq ...[text shortened]... he liquid will lower the center of gravity of the cup, stabilising it. Is that what you mean?[/b]

I'm suggesting there is a curve of maximum accel, obviously when full, it is easier to tip over but I would assume there is a level, maybe associated with the height of the lip, say you have liquid at that level that somewhere around there you have maximum accel before tipping but lower than that, it looks to me like the mass/center of gravity curve would make for a lesser amount of accel before tipping, because there is a lot less fluid inside and it at some point starts to get top heavy again. I hope that makes it clearer.

The coffee will tip only while accelerating, at a constant speed it should remain calm. So if ya wanna go fast build up slowly then don't break. 🙂

Actually, it's quite a complex problem, and is far beyond my meager capacity for maths. I shall get my Girlfriend to try and work it out, she's got a maths degree...

Actually, we need to know how fast your car can accelerate, so we know maximum acceleration.

Originally posted by Feastboy
The coffee will tip only while accelerating, at a constant speed it should remain calm. So if ya wanna go fast build up slowly then don't break. 🙂

Actually, it's quite a complex problem, and is far beyond my meager capacity for maths. I shall get my Girlfriend to try and work it out, she's got a maths degree...

Actually, we need to know how fast your car can accelerate, so we know maximum acceleration.

The best most cars can accel is about 1 g so thats a starting point.

Originally posted by sonhouse
I'm suggesting there is a curve of maximum accel, obviously when full, it is easier to tip over but I would assume there is a level, maybe associated with the height of the lip, say you have liquid at that level that somewhere around there you have maximum accel before tipping but lower than that, it looks to me like the mass/center of gravity curve would m ...[text shortened]... fluid inside and it at some point starts to get top heavy again. I hope that makes it clearer.

No, that is extraordinarily unclear.

Originally posted by AThousandYoung
No, that is extraordinarily unclear.

I guess its a matter of defining topheavy. If the cup is full, it is topheavy, so it doesn't take much force to tip it over, the force coming from the accelerating car. If the cup is empty, it also doesn't take much g force to tip it over because most of the cup mass is above the 4 Cm height of the support lip. So going from an empty cup to full, somewhere the liquid level gives the best resistance to tipping over from the acceleration. The cup has a lid on it to prevent liquid from spilling over the top. So what is the optimum liquid level that allows the most acceleration without tipping over the cup?

Originally posted by sonhouse
I guess its a matter of defining topheavy. If the cup is full, it is topheavy, so it doesn't take much force to tip it over, the force coming from the accelerating car. If the cup is empty, it also doesn't take much g force to tip it over because most of the cup mass is above the 4 Cm height of the support lip. So going from an empty cup to full, somewhere ...[text shortened]... at is the optimum liquid level that allows the most acceleration without tipping over the cup?

why don't you experiment?

Originally posted by sonhouse
I guess its a matter of defining topheavy. If the cup is full, it is topheavy, so it doesn't take much force to tip it over, the force coming from the accelerating car. If the cup is empty, it also doesn't take much g force to tip it over because most of the cup mass is above the 4 Cm height of the support lip. So going from an empty cup to full, somewhere ...[text shortened]... at is the optimum liquid level that allows the most acceleration without tipping over the cup?

OK. Is this what you mean?

Given a loosely lidded 10 cm tall cup of coffee (of shape XXX, such as cylinder, conic section, whatever, and mass YYY) held loosely in place by a 4 cm high cupholder lip (with no corresponding lip on the far side);

How high should the coffee in the cup be for maximum stability? Stability is defined as resistance to toppling when the restraining lip pushes on the cup, accelerating it, because the car itself is accelerating.

If the road is perfectly straight and smooth, and the center of mass of the partially filled cup is lower than 4cm, the cup will never tip, no matter the acceleration.

Bumps will be resisted by mass, so I guess approximately 8 cm high for a cylindrical cup. Call it 7.5 to be safe.

Originally posted by AThousandYoung
If the road is perfectly straight and smooth, and the center of mass of the partially filled cup is lower than 4cm, the cup will never tip, no matter the acceleration.

Bumps will be resisted by mass, so I guess approximately 8 cm high for a cylindrical cup. Call it 7.5 to be safe.u

But the center of mass itself is a variable. If you accelerate, you are moving a liquid, no? So where does the liquid go when accelerated? To the back of the car, restrained by the cup of course. So what does it do in response to that impetus? The back of the liquid rises and so the center of mass also rises. So it may tip even if the liquid when at rest is at the 4 cm level.
Half of the liquid could be above that 4 cm lip with the surface at a 45 degree angle.

Originally posted by sonhouse
But the center of mass itself is a variable. If you accelerate, you are moving a liquid, no? So where does the liquid go when accelerated? To the back of the car, restrained by the cup of course. So what does it do in response to that impetus? The back of the liquid rises and so the center of mass also rises. So it may tip even if the liquid when at rest is ...[text shortened]... m level.
Half of the liquid could be above that 4 cm lip with the surface at a 45 degree angle.

The center of gravity will shift horizontally but not vertically with a horizontal acceleration. The extra coffee in the back is removed from the front. The up-down position of the COG will not change.

I think.

Originally posted by AThousandYoung
The center of gravity will shift horizontally but not vertically with a horizontal acceleration. The extra coffee in the back is removed from the front. The up-down position of the COG will not change.

I think.

Suppose we look at it like this: if we put a vertical separator down the center of the cup and put all the water on the side facing the back of the car. That represents the extreme of the motion the liquid can do when changing from a horizontal level to a fully vertical level. When it is fully upright, half the weight is on the top half of the cup. It would seem the COG HAS to be higher in that case, doesn't it? If so, then the COG goes higher if it is in an inbetween angle like 45 degrees.

Originally posted by sonhouse
Suppose we look at it like this: if we put a vertical separator down the center of the cup and put all the water on the side facing the back of the car. That represents the extreme of the motion the liquid can do when changing from a horizontal level to a fully vertical level. When it is fully upright, half the weight is on the top half of the cup. It would ...[text shortened]... se, doesn't it? If so, then the COG goes higher if it is in an inbetween angle like 45 degrees.

Maybe you're right. I don't know.

Originally posted by AThousandYoung
Maybe you're right. I don't know.

It seems pretty clear, by definition, if there is less mass lower down and more mass further up, the COG follows, going up.
So if the liquid is tilted at a 45 degree angle from acceleration, mass has been removed from the bottom of the cup and now reaches over the 4 cm lip (still in the cup of course) Look at it geometrically. Suppose you have a rectangle 2 units by 2 units. At the 50% line, halfway up, you draw a line across so you cut the area in half and 50% is above the line and 50% is below the line, in area. Now draw a diagonal from lower left to upper right hand corner of the rectangle, now you have a triangle with a 45 degree angle from the lower left corner to the upper right corner and down to the lower right, that exactly bisects the area also. So if you look at that resultant triangle, you see 1/4th of the area above the center line and 3/4 of the area below the center line, so if you use that figure to represent one side of the cup, then 1/4 of the mass will end up above the 50% line so the COG went up by some amount, not sure what yet. If you consider a cube and its also 2 units on a side and it is half filled with liquid then the COG is at the .25 unit up line. If the liquid is held by a vertical spacer so it occupies half the vertical distance, a spacer from top to bottom putting the liquid all the way to the top and still occupies the same volume, the new COG is now one full unit high, or halfway up the container, in that case 50% of the liquid is above the centerline and 50% is below. In the case of the liquid at a 45 degree angle, only 25% of the liquid is above the center line but is ready to spill out because the liquid stops just at the top now, my finger in the air guess is the new COG is at around the 0.33 mark. When just laying in the bottom, the COG would be at the 0.25 mark, referring to our cube now half full. If it was at the 0.33 mark, that would put the COG above the 4 Cm mark of the supporting lip of the cup, which probably would not tip but it would be tending in that direction. Now I used a cubic area, 2X2X2 but the cup is a slight conic, say equivalent to a cylinder about 50 CmX150Cm. If 4 cm of liquid was in that cylinder and now we put all that liquid vertically oriented, then we would have about 2/3 of the liquid above the 4 cm line, with a big resultant increase in COG, which would be represented by one huge accel for sure! But 1 G of accel would seem to me to vector out to a liquid level of about 45 degrees, so it would be another level of analysis to see if it could tip or just what accel could make it tip against that 4 Cm lip. I don't think 1 G would do it.
If 1 G wouldn't do it, then it seems safe to argue the initial level could be higher yet, higher at least than the 4 Cm lip.